How do I connect the end of the extrusion back to the surface a second time?
Using Bridge edge loops to make the mug one connected surface.
Use the Bridge operator to connect the open edge to a created hole in the mug
Use bevel operator to control the contour of handle to mug connection.

The bridge operator about to be executed. Note the hole at top is selected. The open circular edge of the handle is selected. Two holes aligned. Image above.  Please click on the image to see it larger in a separate web page.


The bridge operator finished. Surfaces are connected and ready for smoothing.  Image above

Create a second hole in the destination area.  The vertex count of the hole should match the extruded vertex count.
Use Mesh Loop Tools to change the hole into a circle, this is a time saver. Fine points not discussed here.
Use Bridge edge loops to make the connection
This whole operation could take 88 seconds for someone experienced.  If you take 8 minutes that is wonderful.


The Mug with Subdivision Surface Modifier and Smooth Shading.  No artistry intended.  You can do a better job.  Please improve and experiment.
There are often many way to do similar things in Blender.  You can experiment with extruding from two holes on the mug and having them meet in the middle with Bridge Edge Loops. Or something similar.  This technique follows.

Mug with two extrusions that meet in the middle of the handle.  In the top 3D View is the edit mode before bridge edge loops and finished version. In the bottom 3D view is showing the extrusion which uses the [Bevel] operator. This can be controlled by you for particular contours at the extrusion.  Handle formed with the Spin operator with a positive angle of 89 degrees and negative angle of 89 degrees.
Bridge edge loops is a time saver.  The other time wasting method is to connect each face separately by selecting the 4 vertices and creating a face with
F
Merge is also available.
There is also a way to get very convincing placement of vertices and lighting information using  two different meshes.  That can wait for another day.